
where we take perturbation limit \(|q_xg^{\prime }|\le \sqrt{4\pi }\) , and choose \(M_1 < M_2\) , and take \(m_{\chi _\ell } < 1.2 \times (M_1, m_{\chi _q})\)  for simplicity so that we can
evade contributions from co-annihilation processes. Here we require 1000 GeV\(\le M_{Q^{\prime }}\)  to avoid constraints from direct search of vector-like quarks at the LHC.
For vector-like lepton, constraint \(m_{VLL} \gtrsim 700\)  GeV is given for vector-like lepton decaying into the SM lepton with \(W/Z\)  boson [1]}.
But here we just consider LEP limit of \(m_{VLL} \gtrsim 100\)  GeV [2]} since our vector-like lepton decays into lepton and \(\chi _l\) .
In addition we impose the constraint of 550 GeV \(\lesssim m_{Z^{\prime }}/g^{\prime }\)  which arises from the neutrino trident production [3]}.
 Here we take scalar mixing \(\sin \alpha \)  to be sufficiently small so that scalar boson contribution to oblique parameters can avoid constraints since it is proportional to \(\sin ^2 \alpha \) .
 We find that new scalar contribution to \(S\)  and \(T\)  parameter is sufficiently small for \(\sin \alpha < 0.1\) .
Moreover, taking \(m_E < M_{E,L^{\prime }}\) , we find that constraint from \(S\) - and \(T\) -parameters can be easily avoided where \(S\)  and \(T\)  are found to be less than \(\sim 0.05\)  and \(0.01\) , respectively.
